# Elliptic Partial Differential Equations: Existence and Regularity of Solutions

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##### Info sul corso
Ore del corso
40
Ore dei docenti responsabili
40
Ore di didattica integrativa
0
CFU 6
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## Modalità esame

Prova orale e relazione di seminario

## Docente

Vedi dettagli del docente

## Prerequisiti

Basic notions of measure theory, PDE and Functional Analysis

Master and PhD.

## Programma

1 Some basic facts regarding Sobolev spaces

2 Variational formulation of some PDEs

2.1 Elliptic operators

2.2 Inhomogeneous boundary conditions

2.3 Elliptic systems

2.4 Other variational aspects

3 Lower semicontinuity of integral functionals

4 Regularity Theory

4.1 Nirenberg method

5 Decay estimates for systems with constant coefficients

6 Regularity up to the boundary

7 Interior regularity for nonlinear problems

8 Holder, Morrey and Campanato spaces

9 XIX Hilbert problem and its solution in the two-dimensional case

10 Schauder theory

11 Regularity in L^p spaces

12 Some classical interpolation theorems

13 Lebesgue differentiation theorem

14 Calderòn-Zygmund decomposition

15 The BMO space

16 Stampacchia Interpolation Theorem

17 De Giorgi’s solution of Hilbert’s XIX problem

17.1 The basic estimates

17.2 Some useful tools

17.3 Proof of Holder continuity

18 Regularity for systems

18.1 De Giorgi’s counterexample to regularity for systems

19 Partial regularity for systems

19.1 The first partial regularity result for systems

19.2 Hausdorff measures

19.3 The second partial regularity result for systems

20 Some tools from convex and non-smooth analysis

20.1 Subdifferential of a convex function

20.2 Convex functions and Measure Theory

21 Viscosity solutions

21.1 Basic definitions

21.2 Viscosity solution versus classical solutions

21.3 The distance function

21.4 Maximum principle for semiconvex functions

21.5 Existence and uniqueness results

21.6 H¨older regularity

22 Regularity theory for viscosity solutions

22.1 The Alexandrov-Bakelman-Pucci estimate

22.2 The Harnack Inequality.

## Obiettivi formativi

In the course the basic techniques of the theory of elliptic partial differential equations are presented. Starting from the very classical methods based on energy inequalities and difference quotients, more recent tools, results and concepts are introduced, touching also the theory of viscosity solutions.

## Riferimenti bibliografici

L.C.EVANS. Partial Differential Equations, American Mathematical Society, 1998.

M.GIAQUINTA, L.MARTINAZZI. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Edizioni della Normale, 2005.

L.AMBROSIO, A.CARLOTTO, A.MASSACCESI, Lecture Notes in PDE, Edizioni della Normale

L.C.EVANS. Partial Differential Equations, American Mathematical Society, 1998.

M.GIAQUINTA, L.MARTINAZZI. An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Edizioni della Normale, 2005.